An equation that models the position of the object at time t is:
s(t) = -2cos(2πt/5).
How to interpret the trigonometric graph?The general form for the equation that will model a wave is:
±a (sin/cos) (2π(x - p)/T)
where:
a is the amplitude
p is the phase shift
T is the period.
The ± will become +ve provided that the graph starts in the positive direction, and the will become -ve provided it starts in the negative direction.
The (sin/cos) will become sine provided the graph starts at 0 before it is being shifted. Then, it becomes cosine provided that the graph starts at the amplitude.
In this case, our graph begins at negative, and the at the amplitude that has no phase shift, the ±ve will become -ve, (sin/cos) will now become cos, and p will become zero. Plugging in the values that were given in the problem, we see that a = 2 and T = 5.
Thus, this equation is: s(t) = -2cos(2πt/5).
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Which expression is equivalent to "9 more than the quotient of x and 5
The required expression is (x / 5) + 9
Given that we have to build an equation for the statement "9 more than the quotient of x and 5,
So,
This expression represents the quotient of x divided by 5, and then adding 9 to the result.
Therefore,
"9 more than the quotient of x and 5" can be written mathematically as:
(x / 5) + 9
Hence the required expression is (x / 5) + 9
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The trapezoids below are similar. Find the two missing side lengths.
Answer:
y = 15, x = 4
Step-by-step explanation:
Hope this helped!
Answer:
y+15x=4
Step-by-step explanation:
6.258 rounded to the nearest hundredth
Answer:
6.26
Step-by-step explanation:
Integer Part: 6
Fractional Part: 258
If the last digit in the fractional part of 6.258 is less than 5, then simply remove the last the digit of the fractional part.
If the last digit in the fractional part of 6.258 is 5 or more and the second digit in the fractional part is less than 9, then add 1 to the second digit of the fractional part and remove the third digit.
If the last digit in the fractional part of 6.258 is 5 or more and the second digit in the fractional part is 9, and the first digit in the fractional part is less than 9, then add 1 to the first digit in fractional part and make the second digit in fractional part 0. Then remove the third digit.
If the last digit in the fractional part of 6.258 is 5 or more and the second digit in the fractional part is 9, and the first digit in the fractional part is 9, then add 1 to the integer part and make the fractional part 00.
With 6.258, rule B applies and 6.258 rounded to the nearest hundredth is 2.26
please help me! math stuff
Answer:
C.
Step-by-step explanation:
64+56=120
189-120=C. $69
Find the 14th term of the arithmetic sequence x-4, 7x-9, 13x-14, ...
Answer:
14x-9
Step-by-step explanation:
What is the measure of XY?
A. 127°
B. 53°
C. 37°
D. 90°
Answer:
it is B. 53
Step-by-step explanation:
Answer:
The answer is 53
Step-by-step explanation:
I learned about it.
A standard dice is tossed twice. What is the probability of obtaining exactly one 5? Express your answer as a common fraction.
Answer:
5/18
Step-by-step explanation:
There are a couple of ways to look at this.
1) If you make a matrix of all possibilities, you find there are 36 possible outcomes from the roll of a die twice. (That is the same number as for rolling two dice once.) Of those 36 outcomes, 10 are outcomes in which a 5 shows exactly once: (5, 1), (5, 2), (5, 3), (5, 4), (5, 6), (1, 5), (2, 5), (3, 5), (4, 5), (6, 5).
The probability of obtaining exactly one 5 is 10/36 = 5/18.
__
2) As listed above, there are two ways to get exactly one 5 in two rolls:
(5 on the first, non-5 on the second) or (non-5 on the first, 5 on the second)
When the rolls are independent, as we assume here, the probability of a certain sequence is the product of the probabilities of the events in that sequence.
P(5, non-5) = (1/6)(5/6) = 5/36
P(non-5, 5) = (5/6)(1/6) = 5/36
The probability of obtaining either event is the sum of their individual probabilities:
P({5, 5'} or {5', 5}) = 5/36 +5/36 = 10/36 = 5/18
__
The probability of obtaining exactly one 5 in two rolls of a die is 5/18.
Answer:
S= (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
For this case the size of the sample space is 36 now we count the number of pairs with exactly one 5 and we have:
(1,5), (2,5), (3,5), (4,5), (6,5), (5,6), (5,4), (5,3), (5,2), (5,1)
And then the probability would be:
\( p=\frac{10}{36}= \frac{5}{18}\)
Step-by-step explanation:
For this case w ehave the following sample space:
S= (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
For this case the size of the sample space is 36 now we count the number of pairs with exactly one 5 and we have:
(1,5), (2,5), (3,5), (4,5), (6,5), (5,6), (5,4), (5,3), (5,2), (5,1)
And then the probability would be:
\( p=\frac{10}{36}= \frac{5}{18}\)
PLEASE HELP FAST, FINAL IS TODAY!!‼️WILL GIVE BRAINLIEST‼️
The distance of the fish from the hook is: 21 ft
How to find the length of similar triangles?Two triangles are similar if their corresponding side proportions are the same and their corresponding pairs of angles are the same. When two or more figures have the same shape but different sizes, such objects are called similar figures.
Thus, using the concept of similar triangles, we can say that:
Let x be the horizontal base length from the fish to the hook and as such, we can say that:
3/5 = 12.6/x
x = (12.6 * 5)/3
x = 21 ft
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1.
(03.03 MC)
A biologist is studying the growth of a particular species of algae. She writes the following equation to show the radius of the algae, f(d), in mm, after d days:
f(d) = 11(1.01)d
Part A: When the biologist concluded her study, the radius of the algae was approximately 11.79 mm. What is a reasonable domain to plot the growth function? (4 points)
Part B: What does the y-intercept of the graph of the function f(d) represent? (2 points)
Part C: What is the average rate of change of the function f(d) from d = 2 to d = 7, and what does it represent? (4 points)
Answer:
sept one
Step-by-step explanation:
Pls help me for 20 points
Answer: first one a+3
next one 3(a+3)
next one (a+3)-3
next one a*3
part b
D divided by 4
D-4
4 divide by D
4-D
Step-by-step explanation:
what’s the correct radical form of b^1/5
The correct radical form of b^1/5 is 5^√b.
What is the radical form?Square root and nth roots are represented by the symbol "radical," which. a square root is a component of a radical expression, which is an expression.
A number's or an algebraic expression's simplest radical form is referred to as this. When a number or algebraic expression contains no elements that are perfect nth powers under the radical, it is said to have an nth root and is said to be in its simplest radical form.
When a number or algebraic expression contains no elements that are perfect nth powers under the radical, it is said to have an nth root and is said to be in its simplest radical form.
Explanation:
Convert to radical form using the formula
a^x/n=n^√a^x
5^√b.
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The brain volumes (cm cubed) of 20 brains have a mean of 1162.8 cm cubed and a standard deviation of 127.1 cm cubed. Use the given standard deviation and the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such data, would a brain volume of 1377.0 cm cubed be significantly high?
Answer:
- The limits are 908.6 cm^3 and 1417cm^3
- 1337.0cm^3 is in between the limits
Step-by-step explanation:
To determine the limit by taking into account the range rule of thumb, you use the fact that the limits are given by the mean plus and minus twice the standard deviation, that is:
\(\overline{x}\pm 2\sigma\) (1)
\(\overline{x}\): mean of brain volume = 1162.8 cm^3
σ: standard deviation = 127.1 cm^3
You replace the values of the parameters in the equation (1):
\(1162.8cm^3+2(127.1cm^3)=1417cm^3\\\\1162.8cm^3-2(127.1cm^3)=908.6cm^3\)
Limits = (1417 , 908.6)
The limits are 908.6 cm^3 and 1417cm^3
1337.0cm^3 is in between the limits calculated above.
The perimeter of the triangle shown is 225 feet, find the length of each side
X feet = How many Feet?
5x feet = how many feet?
(6x - 3) feet = for many feet?
Answer:
I have solved it and attached in the explanation.
Step-by-step explanation:
Find the sum and difference of 5 digit greatest and smallest numners.
Answer:
5 digit greatest no: is 99999
Smallest is 10000
Step-by-step explanation:
Sum=
99999+10000
=1,09,999
Subtraction=
99999-10000
=89,999
how much is the scale factor if you dilate it until its area is 36?
Answer:
1/32xp
Step-by-step explanation:
but not sure was there a picture to go with this
To purchase $14,500 worth of restaurant equipment for her business, Debra made a down payment of $1300 and took out a business loan for the rest. After 2 years of paying monthly payments of $585.04, she finally paid off the loan.
(a) What was the total amount Debra ended up paying for the equipment (including the down payment and monthly payments)?
(b) How much interest did Debra pay on the loan?
The total amount Debra ended up paying for the equipment was $28,541.60 and the amount of interest Debra paid on the loan was $14,041.60
(a) To find the total amount Debra ended up paying for the equipment (including the down payment and monthly payments), we need to add the down payment to the total amount of the loan, and then add the total amount of the monthly payments made over the two years.
Total amount of the loan = $14,500 - $1,300 (down payment) = $13,200
Total amount paid = Down payment + Total amount of the loan + Total amount of monthly payments
Total amount paid = $1,300 + $13,200 + ($585.04 x 24) [since there are 24 monthly payments in 2 years]
Total amount paid = $1,300 + $13,200 + $14,041.60
Total amount paid = $28,541.60
Therefore, the total amount Debra ended up paying for the equipment (including the down payment and monthly payments) was $28,541.60.
(b) To find the amount of interest paid on the loan, we need to subtract the total amount borrowed from the total amount paid, and then subtract the down payment. This will give us the total amount of interest paid over the two years.
Total interest paid = Total amount paid - Total amount borrowed - Down payment
Total interest paid = $28,541.60 - $13,200 - $1,300
Total interest paid = $14,041.60
Therefore, the amount of interest Debra paid on the loan was $14,041.60.
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If f(x)= 5x + 40, what is f(x) when x = -5?
Answer:
f(x) = 15
Step-by-step explanation:
"If f(x)= 5x + 40, what is f(x) when x = -5?"
Substitute x for -5:
f(x) = 5x + 40
f(x) = 5(-5) + 40
f(x) = -25 + 40
f(x) = 15
(-2) (4+6)+(-2) 6 / (-2) (4-1) simplified
The simplified form of the expression is \(-32\).
To simplify the expression, we can perform the calculations written below step by step:
\(\(\frac{{-2(4+6)+(-2)6}}{{-2(4-1)}}\)\)
We follow the order of operations (PEMDAS/BODMAS):
Step 1: Simplify within parentheses:
\(\(4+6 = 10\)\).
Step 2: Perform multiplications and divisions from left to right:
\(\(-2(10) = -20\) and \(-2(4-1) = -2(3) = -6\)\).
Step 3: Evaluate the remaining additions and subtractions:
\(\(-20 + (-2) \cdot 6 = -20 - 12 = -32\)\).
Therefore, the simplified form of the expression \(\(\frac{{-2(4+6)+(-2)6}}{{-2(4-1)}}\) is \(-32\).\)
When simplifying an expression, several factors need consideration. First, apply the order of operations correctly, respecting parentheses and exponents. Next, combine like terms by adding or subtracting them. Distribute and simplify within parentheses or brackets as needed. Pay attention to negative signs and ensure their proper placement.
Finally, review the simplified expression to ensure accuracy and validity within the given context.
Note: The complete question is:
\(\(\frac{{-2(4+6)+(-2)6}}{{-2(4-1)}}\)\), calculate the simplified form of this expression.
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Please help me understand this handwriting can some pls re write and label I DONT UNDERSTAND CURSIVE :(
Answer:
7:01
29%
Doyan of low
Shiman 1-R-1
Flous 1-1 Gambe 1-4-20
12 dual purpose
Martins 1-2-3 Flores 1-2, 1-3, 14
Gamba 2-1-5
Custand 1-4, 2-1
Conditions " humanitarian groups Salahiel 1-1-1, 1-1-2, 1-2-3 1-R-5, 2-L-2, 2-L-3, 2-14, 2-1-5, 2-RI
Flores 1-30
Munting 1-2-4
Gumbon 1-1-3 CDP 1-1-3
Status oppose ending 742 Montoya 1-1, 1-2, 1-3
federal courts challenge
Gamboa 1-R-2 Custom 1-7
CDP 1-L-36, 1-2-4
de Vogue 1-2-1, 2-1-5, 2-L-6
7 expulsions at boders
root causes textul Amer.
Cop 1-L-34, 1-1-2, 1-R-4
Border Report 1-2
CDP 1-R-2, 1-R-3 Montoya 1-8
Custarda 2-2
=
CLOSE
Hopes this helps :)
i need an answer!! (marking as brainliest)
Answer: I hope this helps.
Step-by-step explanation:
Which of the following is another way to label Plane J?
Select one: Plane h Plane
ABD Plane EFG Plane ADF
Check
Plane ADF is another way to label Plane J.
In order to determine which of the given options is another way to label Plane J, we need to first identify the key characteristics of Plane J.
These characteristics are its points, lines, and planes that it contains. We can then use these characteristics to compare and match them with the given options.
Here are the key characteristics of Plane J: Points: J, A, D Lines: JA, JD, AD Planes: Plane J Now let's examine each option to see which one matches the characteristics of Plane J: Option A: Plane h This option is not a match since Plane h is not one of the planes that contain the points J, A, and D.
Option B: Plane ABD This option is not a match since Plane ABD contains points A, B, and D but not point J.
Option C: Plane EFG This option is not a match since Plane EFG contains points E, F, and G but not point J.
Option D: Plane ADF This option is a match since Plane ADF contains points A, D, and F which are all points that are contained in Plane J.
Therefore, Plane ADF is another way to label Plane J.
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Marriage Prospects Data released by the Census Bureau in 1986 indicated the likelihood that never-married women would eventually marry. The data indicated that the older the woman, the less the likelihood of marriage. Specifically, two statistics indicated that women who were 45 and never-married had an 18 percent chance of marriage and women 25 years old had a 78 percent chance of marriage. Assume that a linear fit to these two data points provides a reasonable approximation for the function p=f(a), where p equals the probability of marriage and a equals the age of a never- married woman.
(a) Determine the linear function p=f(a).
(b) Interpret the slope and p intercept.
(c) Do the values in part b seem reasonable?
(d) If the restricted domain on this function is 20 sa s 50, determine f(20), f(30), f(40), and f(50).
Answer:
a. \(f(a) = -0.03a +1.53\)
b. See Explanation
c. The slope is reasonable but the p intercept is not
d. \(f(20) = 93\%\) \(f(30) = 63\%\) \(f(40) = 33\%\) \(f(50) = 3\%\)
Step-by-step explanation:
Given
\(a = age\)
\(p = probability\ of\ marriage\)
\(a = 45\) when \(p = 18\%\)
\(a = 25\) when \(p = 78\%\)
Solving (a): The linear function
We start by calculating the slope, m
\(m = \frac{p_2 - p_1}{a_2 - a_1}\)
\(m = \frac{78\% - 18\%}{25- 45}\)
\(m = \frac{60\%}{-20}\)
\(m = -3\%\)
\(m = -0.03\)
The function is then calculated as follows
\(p - p_1 = m(a - a_1)\)
This gives:
\(p - 18\% = -0.03(a - 45)\)
\(p - 0.18 = -0.03(a - 45)\)
\(p - 0.18 = -0.03a +1.35\)
Solve for p
\(p= -0.03a +1.35+0.18\)
\(p= -0.03a +1.53\)
Hence,
\(f(a) = -0.03a +1.53\)
Solving (b): Interpret the slope and the p intercept
The slope is calculated as:
\(m = -0.03\)
And it implies that, there is a 3% reduction in change of getting older as women get older
The p intercept implies that, there is a 1.53 chance for 0 years old female child to get married.
Solving (c): Is (b) reasonable
The slope is reasonable.
However, the p intercept is not because of the age of the woman
Solving (d): Determine f(20), f(30), f(40), f(50)
We have that:
\(f(a) = -0.03a +1.53\)
\(f(20) = -0.03 * 20 + 1.53\)
\(f(20) = -0.6 + 1.53\)
\(f(20) = 0.93\)
\(f(20) = 93\%\)
\(f(30) = -0.03 * 30 + 1.53\)
\(f(30) = -0.9 + 1.53\)
\(f(30) = 0.63\)
\(f(30) = 63\%\)
\(f(40) = -0.03 * 40 + 1.53\)
\(f(40) = -1.2 + 1.53\)
\(f(40) = 0.33\)
\(f(40) = 33\%\)
\(f(50) = -0.03 * 50 + 1.53\)
\(f(50) = -1.5 + 1.53\)
\(f(50) = 0.03\)
\(f(50) = 3\%\)
Three pizza are shared equally among 12 people.what fraction of a pizza will each person get?
A
4/1
B
3/1
Answer:
d) 1/12
Step-by-step explanation:
Which is equivalent to the expression?
(3x - 14)^2
a. 9x^2 - 84x + 196
b. 9x^2 + 84x + 196
c. 9x^2 + 196
d. 9x^2 - 196
Answer:
A
Step-by-step explanation:
(3x - 14)^2
9x^2 - 42x -42x +196
9x^2 - 84 +196 is the correct answer
3 (1 2t) = 3t + 5 !!
Answer:
2/3
Step-by-step explanation:
3+6t=3t+5
like terms together
6t-3t=5-3
3t=2
t=2/3
A production facility employs 10 workers on the day shift, 8 workers on the swing shift, and 6 workers on the graveyard shift. A quality control consultant is to select 4 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 4 workers has the same chance of being selected as does any other group (drawing 4 slips without replacement from among 24).
(a) How many selections result in all 4 workers coming from the day shift? What is the probability that all 4 selected workers will be from the day shift? (Round your answer to four decimal places.)
(b) What is the probability that all 4 selected workers will be from the same shift? (Round your answer to four decimal places.)
(c) What is the probability that at least two different shifts will be represented among the selected workers? (Round your answer to four decimal places.)
(d) What is the probability that at least one of the shifts will be unrepresented in the sample of workers? (Round your answer to four decimal places.)
The probability that all 4 selected workers will be from the day shift is, = 0.0198
The probability that all 4 selected workers will be from the same shift is = 0.0278
The probability that at least two different shifts will be represented among the selected workers is = 0.9722
The probability that at least one of the shifts will be unrepresented in the sample of workers is P(A∪B∪C) = 0.5257
To solve this question properly, we will need to make use of the concept of combination along with set theory.
What is Combination?In mathematical concept, Combination is the grouping of subsets from a set without taking the order of selection into consideration.
The formula for calculating combination can be expressed as:
\(\mathbf{(^n _r) =\dfrac{n!}{r!(n-r)! }}\)
From the parameters given:
Workers employed on the day shift = 10Workers on swing shift = 8Workers on graveyard shift = 6A quality control consultant is to select 4 of these workers for in-depth interviews:
Using the expression for calculating combination:
(a)
The number of selections results in all 4 workers coming from the day shift is :
\(\mathbf{(^n _r) = (^{10} _4)}\)
\(\mathbf{=\dfrac{(10!)}{4!(10-4)!}}\)
= 210
The probability that all 5 selected workers will be from the day shift is,
\(\begin{array}{c}\\P\left( {{\rm{all \ 4 \ selected \ workers\ will \ be \ from \ the \ day \ shift}}} \right) = \dfrac{{\left( \begin{array}{l}\\10\\\\4\\\end{array} \right)}}{{\left( \begin{array}{l}\\24\\\\4\\\end{array} \right)}}\\\end{array}\)
\(\mathbf{= \dfrac{210}{10626}} \\ \\ \\ \mathbf{= 0.0198}\)
(b) The probability that all 4 selected workers will be from the same shift is calculated as follows:
P( all 4 selected workers will be) \(\mathbf{= \dfrac{ \Big(^{10}_4\Big) }{\Big(^{24}_4\Big)}+\dfrac{ \Big(^{8}_4\Big) }{\Big(^{24}_4\Big)} + \dfrac{ \Big(^{6}_4\Big) }{\Big(^{24}_4\Big)}}\)
where;
\(\mathbf{\Big(^{8}_4\Big) = \dfrac{8!}{4!(8-4)!} = 70}\)
\(\mathbf{\Big(^{6}_4\Big) = \dfrac{6!}{4!(6-4)!} = 15}\)
P( all 4 selected workers is:)
\(\mathbf{=\dfrac{210+70+15}{10626}}\)
The probability that all 4 selected workers will be from the same shift is = 0.0278
(c)
The probability that at least two different shifts will be represented among the selected workers can be computed as:
\(= 1-\dfrac{ (^{10}_4) }{(^{24}_4)}+\dfrac{ (^{8}_4) }{(^{24}_4)} + \dfrac{ (^{6}_4) }{(^{24}_4)}\)
\(=1 - \dfrac{210+70+15}{10626}\)
= 1 - 0.0278
= 0.9722
The probability that at least two different shifts will be represented among the selected workers is = 0.9722
(d)
The probability that at least one of the shifts will be unrepresented in the sample of workers is:
\(P(AUBUC) = \dfrac{(^{6+8}_4)}{(^{24}_4)}+ \dfrac{(^{10+6}_4)}{(^{24}_4)}+ \dfrac{(^{10+8}_4)}{(^{24}_4)}- \dfrac{(^{6}_4)}{(^{24}_4)}-\dfrac{(^{8}_4)}{(^{24}_4)}-\dfrac{(^{10}_4)}{(^{24}_4)}+0\)
\(P(AUBUC) = \dfrac{(^{14}_4)}{(^{24}_4)}+ \dfrac{(^{16}_4)}{(^{24}_4)}+ \dfrac{(^{18}_4)}{(^{24}_4)}- \dfrac{(^{6}_4)}{(^{24}_4)}-\dfrac{(^{8}_4)}{(^{24}_4)}-\dfrac{(^{10}_4)}{(^{24}_4)}+0\)
\(P(AUBUC) = \dfrac{1001}{10626}+ \dfrac{1820}{10626}+ \dfrac{3060}{10626}-\dfrac{15}{10626}-\dfrac{70}{10626}-\dfrac{210}{10626} +0\)
The probability that at least one of the shifts will be unrepresented in the sample of workers is P(A∪B∪C) = 0.5257
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The
ratio is traditionally used to measure a company's liquidity.
Answer:
Current Ratio
Step-by-step explanation:
The current ratio is a liquidity ratio that measures how able a company is to pay short-term obligations, or current liabilities, with its current assets.
The formula is \(Current\:Ratio=\frac{Current\:Assets}{Current\:Liabilities}\)
An online customer service department estimates that about 15 percent of callers have to wait more than 8 minutes to have their calls answered by a person. The department conducted a simulation of 1,000 trials to estimate the probabilities that a certain number of callers out of the next 10 callers will have to wait more than 8 minutes to have their calls answered. The simulation is shown in the following histogram.Based on the simulation, what is the probability that at most 2 of the next 10 callers will have to wait more than 8 minutes to have their calls answered?
The probability that at most 2 of the next 10 callers have to wait more than 8 minutes is the sum of the probabilities of 0, 1, or 2 callers having to wait for more than 8 minutes which is 0.810
The probability that at most 2 of the next 10 callers having to wait can be defined using the expression :
P(X ≤ 2) = P(0) + P(1) + P(2)
Using the individual probabilities given by the histogram attached :
P(X ≤ 2) = 0.181 + 0.345 + 0.284
P(X ≤ 2) = 0.810
Therefore, the probability that at most 2 callers have to wait more than 8 minutes is 0.810
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Find the values of x and y if:
4x - 2y= -4 and y = 2/3x + 8/3
Answer: x = 3/4; y = 19/6
Step-by-step explanation:
plug in y = 2/3x + 8/3 into 4x - 2y = -4
4x - 2(2/3x + 8/3) = -4
4x - 4/3x - 18/3 = -4
12/3x - 4/3x = -12/3 + 18/3
8/3x = 6/3
8/3x = 2
x = 2(3/8)
x = 6/8 = 3/4
plug the x value into y = 2/3x + 8/3
y = 2/3(3/4) + 8/3
y = 1/2 + 8/3
y = 3/6 + 16/6
y = 19/6
What is y in the first line if the line passing though (-1,y) and (1,0) perpendicular to the line passing (8,4) and (-1,1) ?
Answer:
\(6\).
Step-by-step explanation:
A line that goes through \((x_{0},\, y_{0})\) and \((x_{1},\, y_{1})\) where \(x_{0} \ne x_{1}\) would have a slope of \(m = (x_{1} - x_{0}) / (y_{1} - y_{0})\).
The slope of the line that goes through \((8,\, 4)\) and \((-1,\, 1)\) would thus be:
\(\begin{aligned}m_{2} &= \frac{1 - 4}{(-1) - 8} \\ &= \frac{(-3)}{(-9)} \\ &= \frac{1}{3}\end{aligned}\).
Two lines in a cartesian plane are perpendicular to one another if and only if the product of their slopes is \((-1)\).
Thus, if \(m_{1}\) and \(m_{2}\) denote the slope of the first and second lines in this question, \(m_{1}\, m_{2} = (-1)\) since the two lines are perpendicular to one another. Since \(m_{2} = (1/3)\), the slope of the first line would be:
\(\begin{aligned} m_{1} &= \frac{(-1)}{m_{2}} \\ &= \frac{(-1)}{(1/3)} \\ &= (-3)\end{aligned}\).
Given that the first line goes through the point \((1,\, 0)\), the point-slope equation of that line would be:
\((y - 0) = (-3)\, (x - 1)\).
\(y = -3\, x + 3\).
Substitute in \(x = (-1)\) to find the \(y\)-coordinate of the point in question:
\(y = -3\times (-1) + 3 = 6\).